This manuscript presents the Phase-Resonant Topological Particle Model, a conceptual framework in which elementary particles are interpreted as localized toroidal configurations of a nonlinear complex phase field. The work develops a minimal scalar field model with a toroidal ansatz, a reduced energy functional, a stationary-radius estimate, and a heuristic mass interpretation based on minimized field energy. It also discusses possible topological quantum numbers, possible fractional charge structures, possible SU(3)-like organization of low-lying sectors, and possible Hopf- or Chern-Simons-type extensions. The manuscript should be regarded as an exploratory geometric and topological research framework rather than a complete particle theory or a direct alternative to the Standard Model.
Oleksandr Kudriavtsev (Sun,) studied this question.